Transcript Slide 1

Experimental design
Based on Chapter 2 of D. Heath (1995). An
Introduction to Experimental Design and
Statistics for Biology. CRC Press.
Four critical features of experimental design
Hurlbert 1984
•
•
•
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Controls
Randomization
Replication
Interspersion
Possible explanations?
Research hypothesis
(or hypotheses)
The design of a experiment
• Factor: humidity
• Variable: direction
Removing other possible effects
• Dealing with bias
Other design issues
• Number of woodlice
• Which woodlice
• They must be representative of the
population of reference
Confounding factors
Independent observations
Analysis
• Null hypothesis:
Probability of damp turn = 0.5
• Alternative hypothesis:
Probability of damp turn = 0.5
Expected frequencies for four trails
damp
dry
dry
dry
dry
damp
damp
dry
dry
damp
dry
dry
damp
dry
damp
damp
damp
damp
dry
damp
dry
damp
dry
dry
dry
damp
damp
damp
dry
damp
Example
• Damp*Damp*Damp*Damp
• If order does not matter there is only one way to
obtain four damp turns and the combined
probability (under the assumption of
independence) is 0.5*0.5*0.5*0.5= 0.0625
• Calculate the probability of the other possible
outcomes under the null hypothesis
Exercise
• There are four ways to obtain three damp turns:
Damp*Damp*Damp*Dry
Damp*Damp*Dry*Damp
Damp*Dry*Damp*Damp
Dry*Damp*Damp*Damp
• and the combined probability (under the
assumption of independence) is
0.5*0.5*0.5*0.5= 0.0625 four times = 0.25
• Calculate the probability of the other
possible outcomes under the null hypothesis
Binomial distribution (4 trials)
Under the null hypothesis
Expected frequency
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0
1
2
3
Number of damp turns
4
Distribution under the null hypothesis
(17 trials)
Expected frequency
0.20
0.15
0.10
0.05
0.00
0:17
2:15
4:13
6:11
8:09 10:07 12:05 14:03 16:01
Number of damp:dry turns
What do you conclude if
we observed 14 damp
turns out of 17 ?
Binomial distribution
Rejection region
Rejection region
unlikely
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
likely
unlikely
0.0000+
0.0001+
0.0010+
0.0052+
0.0182=
0.0182+
0.0052+
0.0010+
0.0001+
0.0000=
2.45%
2.45%
2
4
6
8
10
12
Number of damp turns
14
16
Why we start with the null
hypothesis?
The main points
• Use a mathematical model to produce a
sampling distribution of all possible
values of the test statistic assuming that
the null hypothesis is true
• Find the probability associated with a a
particular value occurring in a particular
experiment
• Use the probability to make a decision
about whether a particular result is likely
or unlikely
The Binomial Distribution
Overview
• However, if order is not important, then
P(X) =
n!
 pX  qn – X
X!(n – X)!
n!
where X!(n – X)! is the number of ways to obtain X
successes
in n trials, and n! = n  (n – 1)  (n – 2)  …  2  1